let C be Category; :: thesis: for a, b, c being Object of C
for f being Morphism of a,b
for g being Morphism of b,c st Hom (a,b) <> {} & Hom (b,c) <> {} holds
(g * f) opp = (f opp) * (g opp)
let a, b, c be Object of C; :: thesis: for f being Morphism of a,b
for g being Morphism of b,c st Hom (a,b) <> {} & Hom (b,c) <> {} holds
(g * f) opp = (f opp) * (g opp)
let f be Morphism of a,b; :: thesis: for g being Morphism of b,c st Hom (a,b) <> {} & Hom (b,c) <> {} holds
(g * f) opp = (f opp) * (g opp)
let g be Morphism of b,c; :: thesis: ( Hom (a,b) <> {} & Hom (b,c) <> {} implies (g * f) opp = (f opp) * (g opp) )
assume A1: ( Hom (a,b) <> {} & Hom (b,c) <> {} ) ; :: thesis: (g * f) opp = (f opp) * (g opp)
then A2: g * f = g * f by CAT_1:def 10;
( cod f = b & dom g = b ) by A1, CAT_1:5;
hence (g * f) opp = (f opp) * (g opp) by A2, Th17; :: thesis: verum